Upon being drafted, Bowman returned to the Chiefs and recorded 82 points in 66 games, first in team scoring.[4] His 42 goals tied for fourth in the league.[5] Bowman added a team-leading 20 points in 21 playoff games[6] as the Chiefs captured the Ed Chynoweth Cup as WHL champions. The league title earned the Chiefs a berth in the 2008 Memorial Cup in Kitchener, Ontario. Bowman notched a hat trick in the opening game of the tournament, including the game-tying goal late in the third period of a 5–4 overtime win over the Belleville Bulls.[7] He went on to score in all four games of the tournament, including game-winners against the Kitchener Rangers in the round-robin and final, as the Chiefs went undefeated to capture the Memorial Cup as Canadian major junior champions.[7][8] With six goals in four games,[7][8] Bowman was named to the Memorial Cup All-Star Team.[7]

Following his Memorial Cup performance, the Hurricanes signed him to a three-year, US$2.06 million contract on July 31, 2008.[9] Playing in his third season with the Chiefs in 2008–09, Drayson was named an alternate captain to Justin McCrae along with Seth Compton and Jared Spurgeon.[10] He was named WHL and CHL Player of the Week after recording 12 points in three games for the week ending February 1, 2009.[11] The next month, he earned his second WHL and CHL Player of the Week distinction with an eight-point effort in two games for the week ending March 15, 2009.[12] He finished the season with 47 goals, fourth in the league,[13] and 83 points. He was named to the WHL West Second All-Star Team along with teammate Dustin Tokarski.[14] Bowman and the Chiefs were not, however, able to defend their WHL or CHL titles as they were eliminated in seven games of the second round of the WHL playoffs by the Vancouver Giants.

Upon the completion of Bowman's third WHL season, he was called up by the Hurricanes to travel and practice with the team during their 2009 playoff run.[15] The following season, he was assigned to the Hurricanes' American Hockey League (AHL) affiliate, the Albany River Rats.[16] He scored his first professional goal in his AHL debut with Albany on October 3, 2009, in a 6–3 loss to the Manchester Monarchs.[17] He was recalled from the AHL midway through the season and made his NHL debut on January 16, 2010, a 5–3 loss to the Atlanta Thrashers.[18] After being sent back down to Albany, he received another call-up on March 24 in light of an injury to forward Tuomo Ruutu.[19] He scored his first and second NHL goals both in the first period of his seventh game with the Hurricanes on April 6 in an 8–5 win against the Tampa Bay Lightning.[20]

Bowman made the Hurricanes' roster out of training camp in 2010–11, but was returned to the AHL within a month.[21] With the Hurricanes having changed their minor league affiliate, he joined a new team, the Charlotte Checkers. After recording 30 points (12 goals and 18 assists) over 51 games with the Checkers, he was re-called on March 10, 2011.[21] Spending the remainder of the season with the Hurricanes, he finished 2010–11 with one assist over 23 games, averaging 10 minutes of ice time,[22] in the NHL. Carolina failed to qualify for the playoffs, coming within three points of the eighth and final seed in the Eastern Conference. With his NHL season over, the Hurricanes returned him to the AHL, where the Checkers had qualified for the Calder Cup playoffs. Bowman contributed 8 points (2 goals and 6 assists) over 15 games as the Checkers were eliminated in the Conference Finals by the Binghamton Senators.